Do filters complementive to a given filter form a complete lattice? (Solved)
Let is a set. A filter (on ) is by definition a non-empty set of subsets of such that . Note that unlike some other authors I do not require . I will denote the lattice of all filters (on ) ordered by set inclusion.
Let is some (fixed) filter. Let . Obviously is a bounded lattice.
I will call complementive such filters that:
- ;
- is a complemented element of the lattice .
Conjecture The set of complementive filters ordered by inclusion is a complete lattice.
To this problem was found a counterexample in a MathOverflow question answer.
The counter example with a rewritten proof is also available in Filters on Posets and Generalizations online article.
Bibliography
*Victor Porton. Do filters complementive to a given filter form a complete lattice?
* indicates original appearance(s) of problem.